52 lines
1.5 KiB
Text
52 lines
1.5 KiB
Text
inductive ListSplit {α : Type u} : List α → Type u
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| split l₁ l₂ : ListSplit (l₁ ++ l₂)
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def splitList {α : Type _} : (l : List α) → ListSplit l
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| [] => ListSplit.split [] []
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| h :: t => ListSplit.split [h] t
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theorem Nat.lt_add_left {m n : Nat} : m < n + m := sorry
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theorem Nat.lt_add_right {m n : Nat} : m < m + n := sorry
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def len : List α → Nat
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| [] => 0
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| a :: [] => 1
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| l =>
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match splitList l with
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| ListSplit.split fst snd => len fst + len snd
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termination_by _ l => l.length
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decreasing_by
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simp [measure, id, invImage, InvImage, Nat.lt_wfRel, WellFoundedRelation.rel, sizeOf] <;>
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first
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| apply Nat.lt_add_right
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| apply Nat.lt_add_left
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theorem len_nil : len ([] : List α) = 0 := by
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simp [len]
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-- The `simp [len]` above generated the following equation theorems for len
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#check @len._eq_1
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#check @len._eq_2
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#check @len._eq_3 -- It is conditional, and may be tricky to use.
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theorem len_1 (a : α) : len [a] = 1 := by
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simp [len]
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theorem len_2 (a b : α) (bs : List α) : len (a::b::bs) = 1 + len (b::bs) := by
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conv => lhs; unfold len
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-- The `unfold` tactic above generated the following theorem
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#check @len._unfold
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theorem len_cons (a : α) (as : List α) : len (a::as) = 1 + len as := by
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cases as with
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| nil => simp [len_1, len_nil]
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| cons b bs => simp [len_2]
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theorem listlen : ∀ l : List α, l.length = len l := by
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intro l
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induction l with
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| nil => rfl
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| cons h t ih =>
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simp [List.length, len_cons, ih]
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rw [Nat.add_comm]
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